Composite Fermi Liquid (CFL) Overview

• Composite Fermi Liquid (CFL) is a compressible, gapless state in fractional quantum Hall systems where electrons bind flux quanta to form composite fermions with a hidden Fermi surface.
• It exhibits non-Fermi-liquid behavior due to strong coupling between composite fermions and emergent gauge fields, leading to unique thermodynamic and entanglement properties.
• Microscopic approaches, including the Rezayi–Read construction and Son’s Dirac theory, provide diverse effective descriptions that extend CFL concepts to Chern bands and multilayer systems.

The composite Fermi liquid (CFL) is a compressible, gapless, non-Fermi-liquid state that occurs in fractional quantum Hall systems, especially at even-denominator fillings of a Landau level. In the standard flux-attachment picture, electrons bind vortices to become composite fermions, and these composite fermions can form a Fermi sea. The state is therefore Fermi-liquid-like in having a Fermi surface and Luttinger-counting features, but it is not an ordinary Fermi liquid because the composite fermions are coupled to an emergent gauge field and arise from strong interactions rather than weakly interacting quasiparticles (Voinea et al., 2024).

1. Foundational definition and physical content

At half filling of the lowest Landau level, $\nu=1/2$, the standard interpretation is that electrons bind two flux quanta and form composite fermions. These composite fermions see, on average, zero effective magnetic field and therefore form a gapless Fermi sea rather than an incompressible quantum Hall state. In this sense, the CFL is the canonical compressible metallic state of the half-filled Landau level, but its metallicity is not that of a Landau Fermi liquid because gauge fluctuations remain active at low energy (Barkeshli et al., 2012).

The conventional Halperin–Lee–Read description realizes this structure by attaching statistical flux to electrons. A representative effective Lagrangian is

$$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$

with $f$ the composite fermion, $a_\mu$ the emergent gauge field, and $A_\mu$ the external electromagnetic field (Barkeshli et al., 2012). At mean field, the average emergent flux cancels the physical magnetic field, leaving a composite-fermion Fermi surface.

A distinct but closely related lowest-Landau-level perspective treats the composite fermion as a charge-neutral vortex carrying physical vorticity. In that formulation, the effective theory has no Chern-Simons term for the internal gauge field, and the Fermi surface encloses a Berry phase fixed by the filling fraction,

$\phi_B=-2\pi \nu.$

This lowest-Landau-level viewpoint does not require microscopic particle-hole symmetry, although $\nu=1/2$ is a special case where particle-hole symmetry is exact in the ideal lowest-Landau-level limit (Wang et al., 2016).

2. Microscopic constructions and effective descriptions

The standard variational starting point is the Rezayi–Read form

$$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$

where $\chi_n$ are composite-fermion orbitals in zero field and $\psi_m^\mathrm{L}$ is the Laughlin-Jastrow factor at filling $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$0. For fermionic CFLs, $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$1 is even; for bosonic CFLs, $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$2 is odd. This construction was used in a recent entanglement study for fermionic CFLs at $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$3 and $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$4, and bosonic CFLs at $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$5 and $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$6, in both unprojected and lowest-Landau-level-projected forms (Voinea et al., 2024).

A broader framework is the projective or parton construction

$$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$7

or in continuum notation $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$8, where the charged bosonic sector and neutral fermionic sector are coupled by an emergent $$\mathcal{L} = f^\dagger (\partial_\tau - ia_0 - \mu) f -\frac{1}{2m_b} f^\dagger (\partial - iA - ia)^2 f + \frac{i}{4\pi(2m)} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda + \cdots,$$9 gauge field. When the bosonic sector is in a $f$0 Laughlin state, integrating out the topological charge sector reproduces the CFL; when the boson condenses, the gauge field is Higgsed and the result is an ordinary Landau Fermi liquid (Barkeshli et al., 2012). This construction is useful because it extends beyond ordinary flux attachment and continues to apply in Chern bands without external magnetic field.

At exact half filling, Son’s Dirac composite fermion theory provides a particle-hole-symmetric alternative low-energy description. Away from half filling, a “flux-attached Dirac” hierarchy has been proposed for fermionic CFLs at $f$1 and their particle-hole conjugates. In that construction, the total Berry phase accumulated around a Fermi-surface loop is

$f$2

which numerically supports the idea that a Dirac-like $f$3 singularity survives while additional internal gauge flux attachment produces a uniform background Berry curvature over the Fermi sea (Wang, 2018).

3. Entanglement scaling and charge-fluctuation diagnostics

A recent continuum study revisited the entanglement structure of the CFL using Monte Carlo evaluation of the second Rényi entropy,

$f$4

For the CFL, the study found

$f$5

with a multiplicative enhancement

$f$6

This enhancement was observed at all studied fillings $f$7, in both torus and sphere geometries, and for both projected and unprojected wave functions. Lowest-Landau-level projection was found to have only a very small effect on the total entanglement slope, except for the bosonic $f$8 case, where the projection effect is somewhat stronger. The same work also found that subsystem particle-number fluctuations are strongly suppressed relative to free fermions and obey

$f$9

with a universal corner term

$a_\mu$0

for a region with a sharp corner of opening angle $a_\mu$1. These results were presented as complementary fingerprints of non-Fermi-liquid correlations in CFL states (Voinea et al., 2024).

The entanglement coefficient is, however, a point of active interpretation. A lattice realization of the $a_\mu$2 CFL on the triangular lattice argued that the asymptotic leading prefactor of the $a_\mu$3 term is consistent with the free-fermion Widom value once the second Rényi entropy is decomposed into modulus and sign contributions,

$a_\mu$4

In that analysis, the sign structure carried the long-distance multiplicative logarithm, while the modulus term produced strong finite-size enhancement without changing the asymptotic leading coefficient (Mishmash et al., 2016). Taken together, these works show that the existence, magnitude, and interpretation of entanglement enhancement depend sensitively on geometry, finite-size control, and the diagnostic used.

The same continuum study also considered a “flux-attached” trial state

$a_\mu$5

for which the modulus entropy matches that of free fermions, so any enhancement is shifted entirely into the sign structure (Voinea et al., 2024). This suggests that the relation between entanglement growth and many-body phase structure is not exhausted by Fermi-surface kinematics alone.

4. Berry phase, particle-hole symmetry, and conjugate compressible states

Berry-phase structure is central to modern CFL theory. In one lowest-Landau-level formulation, the composite-fermion Fermi surface encloses the universal Berry phase $a_\mu$6, which yields $a_\mu$7 at $a_\mu$8. This result is tied to the interpretation of the composite fermion as a neutral dipolar vortex, rather than as a charged quasiparticle with a bare Chern-Simons term (Wang et al., 2016).

Microscopic studies refine this picture. For the standard half-filled CFL wave function, the Berry curvature was shown analytically to be uniformly distributed in momentum space, while for the Jain–Kamilla wave function it was shown numerically to have a continuous distribution inside the Fermi sea and to vanish outside. Although the Berry phase around the Fermi circle is $a_\mu$9 in the composite-fermion representation, the conclusion of that study was that the microscopic composite fermion is not a massless Dirac particle (Ji et al., 2019). By contrast, the $A_\mu$0 Berry-curvature study that motivated the flux-attached Dirac hierarchy found a $A_\mu$1 singular contribution at the Fermi-sea center plus a uniform background contribution over the sea, and interpreted this as evidence that a Dirac singularity persists away from $A_\mu$2 in a generalized form (Wang, 2018).

Particle-hole symmetry complicates the half-filled case further. The particle-hole conjugate of the CFL, the Anti-CFL, has been argued to be a distinct phase of matter rather than a mere reformulation of the same state. In that view, the clean half-filled lowest Landau level should be regarded as a setting in which CFL and Anti-CFL compete, with interfaces that permit nontrivial transmutation between “composite electrons” and “composite holes,” and a possible continuous transition controlled by a neutral Dirac fermion coupled to emergent gauge fields (Barkeshli et al., 2015). This makes the phrase “the CFL at half filling” potentially ambiguous unless the particle-hole structure is specified.

5. Geometry, anisotropy, and multilayer structure

The CFL does not simply inherit the geometry of the zero-field electron Fermi surface. For an anisotropic bare band mass at $A_\mu$3, infinite-cylinder iDMRG studies extracted the composite-fermion Fermi contour from singularities in the guiding-center structure factor and found, for Coulomb interactions,

$A_\mu$4

with a power-law fit $A_\mu$5 giving $A_\mu$6. For dipolar interactions $A_\mu$7, the exponent changed to $A_\mu$8, and for a Gaussian interaction the numerics agreed well with Yang’s analytic formula. The mapping from bare anisotropy to composite-fermion anisotropy is therefore interaction-dependent rather than universal (Ippoliti et al., 2017).

More generally, for band anisotropy with discrete rotational symmetry $A_\mu$9, the anisotropy transmitted to the composite fermions decreases rapidly with $\phi_B=-2\pi \nu.$0. DMRG studies found that for $\phi_B=-2\pi \nu.$1 the composite-fermion anisotropy saturates around $\phi_B=-2\pi \nu.$2, while for $\phi_B=-2\pi \nu.$3 it is consistent with $\phi_B=-2\pi \nu.$4 within numerical uncertainty. The explanation is that Landau-level projection strongly suppresses higher-harmonic anisotropy in the effective interaction, so experimentally detectable distortions of the composite Fermi contour are expected to be dominated by the elliptical $\phi_B=-2\pi \nu.$5 channel (Ippoliti et al., 2017).

A complementary result is that even dramatic zero-field Fermi-surface reconstructions do not necessarily survive into the CFL. For rotationally symmetric dispersions, if the lowest-energy Landau level remains $\phi_B=-2\pi \nu.$6, then the $\phi_B=-2\pi \nu.$7 many-body problem is exactly the same as for the usual quadratic dispersion, and the CFL remains circular even if the zero-field Fermi sea is annular or consists of multiple disconnected rings. In non-rotationally symmetric cases, multiple zero-field pockets do not generically imply a multivalley CFL; the composite-fermion Fermi contour can remain single and connected (Ippoliti et al., 2017).

Bilayer systems provide another geometric diagnostic. In the interlayer coherent composite Fermi liquid (ICCFL), interlayer coherence of composite fermions yields a metallic state in which the symmetric gauge sector remains gapless while the antisymmetric sector is Higgsed. In Son’s Dirac formulation, the interlayer exciton inherits momentum-angle dependence from the underlying Berry phase, producing nematic bonding and antibonding composite Fermi seas with hot spots, half-quantum vortices with deconfined $\phi_B=-2\pi \nu.$8 gauge flux, a Lifshitz criticality with $\phi_B=-2\pi \nu.$9, and a Wen–Zee term linking layer imbalance to background curvature (You, 2017).

6. Thermodynamics, instabilities, and nearby phases

Finite-temperature simulations reinforce the non-Fermi-liquid interpretation. Thermal tensor-network calculations on the $\nu=1/2$0 lowest-Landau-level problem found a low-temperature specific heat

$\nu=1/2$1

for sufficiently large systems, in clear contrast to the ordinary Fermi-liquid form $\nu=1/2$2. The same study found that the guiding-center density correlation develops a clear $\nu=1/2$3 circle below about $\nu=1/2$4, while the small-$\nu=1/2$5 behavior remains $\nu=1/2$6. The resulting picture is that the CFL has a hidden Fermi surface but non-Fermi-liquid thermodynamics because the composite fermions are coupled to a dynamical emergent gauge field (Chen et al., 2 Sep 2025).

The CFL is also susceptible to symmetry-breaking instabilities. A field theory of the isotropic-to-nematic transition in the half-filled Landau level starts from the HLR CFL and adds an attractive quadrupolar interaction. The nematic order parameter acts as an effective dynamical metric coupled to the Chern-Simons gauge field, and the resulting critical theory has dynamical exponent $\nu=1/2$7 due to Landau damping. Gauge fluctuations induce a Berry-phase or Hall-viscosity-like term in the nematic action, as well as a Wen–Zee-type coupling, and the theory predicts that nematic disclinations carry electric charge (You et al., 2016).

Bandwidth tuning provides another route out of the CFL. In a parton formulation, increasing the Landau-level bandwidth relative to interaction strength can drive a continuous transition from the CFL to a Landau Fermi liquid by condensing the bosonic charge sector. More generically, the theory predicts an intermediate gapless Mott insulator with a neutral fermion Fermi surface, two crossover temperature scales,

$\nu=1/2$8

a resistivity jump, and vanishing zero-temperature compressibility at the direct critical point (Barkeshli et al., 2012).

Attractive interactions reorganize these possibilities. Near the CFL–Landau-Fermi-liquid transition in a parton theory relevant to rhombohedral graphene, gauge fluctuations suppress pairing strongly enough that the CFL itself is stable to weak attraction, and the nearby Landau Fermi liquid can also remain stable in a finite regime. Under weak attraction, the evolution from CFL to chiral superconductor generically proceeds through an intermediate stable Landau Fermi liquid; under stronger attraction, composite fermions can pair on the CFL side and produce the non-Abelian Moore–Read state, allowing the alternate route CFL $\nu=1/2$9 Moore–Read $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$0 chiral superconductor (Zhang et al., 25 Sep 2025).

7. Zero-field Chern bands, conjugate CFLs, and moiré generalizations

The CFL framework is no longer confined to continuum Landau levels. In twisted bilayer MoTe$$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$1, exact diagonalization and DMRG studies at half filling of the topological valence band found a broad zero-field CFL region centered around twist angle $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$2. The evidence includes a near one-to-one correspondence between finite-size spectra and those of the half-filled lowest-Landau-level Coulomb problem, momentum sectors matching compact composite Fermi sea configurations, featureless electron occupation $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$3 in the realistic interaction regime, and sharp structure in the density correlation at $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$4. The resulting state is a compressible, metallic, non-Fermi-liquid Chern-band analog of the half-filled-Landau-level CFL (Dong et al., 2023).

When a pair of opposite-Chern bands are both half-filled, the natural compressible state is a conjugate composite Fermi liquid (cCFL), namely two CFLs related by time reversal. In one proposed realization, in-plane spin order Higgses the spin gauge field and yields a quantum bad metal with finite zero-temperature longitudinal resistivity satisfying $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$5, together with a modified Wiedemann–Franz law in which the thermal conductivity is proportional to the electrical resistivity rather than the conductivity. The same framework admits proximate superconducting and chiral spin liquid phases generated by inter-valley composite-fermion exciton condensation (Myerson-Jain et al., 2023).

A related construction starts from two decoupled opposite-chirality CFLs and, with strong inter-valley repulsion, produces a vortex spin liquid with fractional quantum spin Hall effect. In that phase, exciton pairing of the composite fermions Higgses the antisymmetric gauge sector, opens a charge gap, and leaves behind neutral and spinless Fermi surfaces of vortices coupled to an emergent $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$6 gauge field. The bulk remains charge insulating but supports the topological response

$$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$7

implying helical charge edge modes (Zhang, 2024).

Pairing in half-filled Chern bands can also lead to gapless topological descendants. In inversion-asymmetric $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$8-symmetric Chern bands, composite fermions may form a superconductor with neutral Bogoliubov Fermi surfaces, producing the composite Bogoliubov Fermi liquid (CBFL). This state is incompressible and Hall-quantized like a paired quantum Hall fluid, but retains metallic $$\Psi_{m}^\mathrm{CFL}(z_1,\ldots,z_N) = \mathrm{Det}\big[\chi_{n}(z_j,z_j^*)\big]\, \psi_m^\mathrm{L}(z_1,\ldots,z_N),$$9-linear specific heat, non-quantized thermal conductance, Landau-damped density fluctuations, and a non-analytic $\chi_n$0 contribution to the equal-time structure factor because the paired composite-fermion spectrum is not fully gapped (Shi et al., 14 Jan 2026).

These zero-field and multiband extensions suggest that “CFL” has become a broader organizing principle for fractionalized metals in topological bands. The common element is not the presence of an external magnetic field, but the combination of a hidden Fermi surface of emergent fermions, strong gauge-field dynamics, and a charged sector whose topology or geometry supports quantum Hall-like fractionalization.